P1^
6
Exercise(^) 6F
EXERCISE 6F 1 Evaluate the following indefinite integrals.
(i) (^) ∫()xx+ 54 d (ii) (^) ∫()xx+ 78 d
(iii) 1
()x 26
x
∫ −
d (iv) (^) ∫ xx−4d
(v) (^) ∫() 31 xx−^3 d (vi) (^) ∫() 52 xx−^6 d
(vii) (^) ∫ 32 ()xx− 45 d (viii) (^) ∫ 42 xx− d
(ix) 4
∫() 8 −x 2
dx (x) 3
21 x
x
∫ −
d
2 Evaluate the following definite integrals.
(i) (^1)
5
∫ x x−^1 d^ (ii)^ ∫^31 ()xx+^13 d^
(iii) (^) ∫−^41 ()xx− 34 d (iv) (^) ∫ 03 () 42 − xx^5 d
(v) (^) ∫ 59 xx− 5 d (vi) (^) ∫ 210 xx− 1 d
3 The graph of y = (x – 2)^3 is shown here.
(i) Evaluate (^) ∫ 24 ()xx− 23 d.
(ii) Without doing any calculations, state
what you think the value of
(^0)
(^223)
∫ ()xx− d^ would^ be.^ Give^ reasons.
(iii) Confirm your answer by carrying out
the integration.
4 The graph of y = (x – 1)^4 – 1 is shown here.
(i) Find the area of the shaded region A by evaluating (^) ∫−^01 ()()xx−− 114 d.
(ii) Find the area of the shaded region B by evaluating an appropriate integral.
(iii) Write down the area of the total shaded region.
(iv) Why could you not just evaluate (^) ∫−^21 ()()xx−− 114 d to find the total area?
y
O^24 x
y = (x – 2)^3
y
O B 2 x
A
y = (x – 1)^4 – 1